Rearrange exercises and clarify tasks

odlgroup · Dec 4, 2017 · 1d6ed8b · 1d6ed8b
1 parent d7803ba
commit 1d6ed8b
Showing 1 changed file with 48 additions and 24 deletions.
diff --git a/code/Exercises_Part1.ipynb b/code/Exercises_Part1.ipynb
@@ -19,27 +19,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## 1. Tomography\n",
-    "\n",
-    "Instead of denoising, solve the TV-regularized tomography problem\n",
-    "\n",
-    "$$\n",
-    "    \\min_{x \\geq 0} \\left[ \\| A x - y \\|_2^2 + \\alpha \\| \\nabla x \\|_1 \\right],\n",
-    "$$\n",
-    "\n",
-    "where $A$ is the ray transform. You may pick your favorite acquisition geometry (2D, 3D, parallel beam, cone beam, ...). Play with different solvers, regularizers and data fit terms (see e.g. Exercises 2 and 3).\n",
-    "\n",
-    "### Good to know\n",
-    "\n",
-    "- Find out how to set up the ray transform as ODL operator. You should use the ASTRA toolbox as backend if possible.\n",
-    "- The data fit $\\|\\cdot -y\\|_2^2$ is now defined on a different space than before, and also $y$ is no longer an element of the reconstruction space $X$. Make sure you understand the details."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## 2. Smooth solvers\n",
+    "## 1. Smooth solvers\n",
     "\n",
     "ODL also has a number of numerical solvers for smooth problems, most of them first-order methods, i.e., methods that involve the gradient of the cost function. The [LBFGS](https://github.com/odlgroup/odl/blob/ad32a286b69f34260d4428d7282b4058ed2e2603/odl/solvers/smooth/newton.py#L247-L487) method is the limited memory variant of the popular [BFGS algorithm](https://en.wikipedia.org/wiki/Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm) for smooth optimization.\n",
     "\n",
@@ -110,6 +90,11 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
+    "### Tasks\n",
+    "\n",
+    "- Set up the smooth version of the TV minimization problem using the Huber function.\n",
+    "- Solve it with a smooth solver, using, e.g., BFGS.\n",
+    "\n",
     "### Good to know\n",
     "\n",
     "- You may think that this is a lot of work. It isn't! That's because the Huber function is the **Moreau envelope** of the 1-norm. You don't need to know what this means exactly -- the important property that you will apply is\n",
@@ -129,7 +114,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## 3. Other goal functions\n",
+    "## 2. Other goal functions\n",
     "\n",
     "So far we have considered $\\|\\cdot -y\\|_2^2$ as data fit and $\\|\\nabla \\cdot\\|_1$ as regularization term. Both can be changed, of course. \n",
     "\n",
@@ -148,7 +133,41 @@
     "- $\\|W \\cdot\\|_1$ with the wavelet decomposition operator $W$,\n",
     "- ...\n",
     "\n",
-    "Experiment with these alternatives."
+    "### Tasks\n",
+    "\n",
+    "- Generate an image that is corrupted with Poisson noise instead of white noise.\n",
+    "- Use KL as data fit in the TV denoising problem to remove the Poisson noise.\n",
+    "- Do the same with salt-and-pepper noise and $L^1$ data fit.\n",
+    "\n",
+    "### Good to know\n",
+    "\n",
+    "- All these noise variants are implemented in ODL.\n",
+    "- The KL divergence is differentiable, so you can use a smooth solver for the Huber-TV variant if you like."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## 3. Tomography\n",
+    "\n",
+    "Instead of denoising, solve the TV-regularized tomography problem\n",
+    "\n",
+    "$$\n",
+    "    \\min_{x \\geq 0} \\left[ \\| A x - y \\|_2^2 + \\alpha \\| \\nabla x \\|_1 \\right],\n",
+    "$$\n",
+    "\n",
+    "where $A$ is the ray transform. You may pick your favorite acquisition geometry (2D, 3D, parallel beam, cone beam, ...). Play with different solvers, regularizers and data fit terms (see e.g. Exercises 2 and 3).\n",
+    "\n",
+    "### Tasks\n",
+    "1. Solve the TV-regularized tomography problem.\n",
+    "2. Use at least one non-smooth and one smooth solver (see exercise 1 for the latter).\n",
+    "3. Add Poisson noise instead of white noise, and use the KL divergence for regularization (see exercise 2).\n",
+    "\n",
+    "### Good to know\n",
+    "\n",
+    "- Find out how to set up the ray transform as ODL operator. You should use the ASTRA toolbox as backend if possible.\n",
+    "- The data fit $\\|\\cdot -y\\|_2^2$ is now defined on a different space than before, and also $y$ is no longer an element of the reconstruction space $X$. Make sure you understand the details."
    ]
   },
   {
@@ -197,7 +216,12 @@
     "\n",
     "The Kaczmarz method then successively updates $x$ using one block at a time, and repeating the outer loop a given number of times. (The [Wikipedia page](https://en.wikipedia.org/wiki/Kaczmarz_method) explains the basic form of the algorithm.)\n",
     "\n",
-    "Use [`odl.solvers.kaczmarz`](https://github.com/odlgroup/odl/blob/ad32a286b69f34260d4428d7282b4058ed2e2603/odl/solvers/iterative/iterative.py#L387-L517) to solve a tomography problem by splitting along the angles. Experiment a bit with blocking schemes (block-sequential, interlaced, maximizing inter-block angle distance etc.).\n",
+    "### Tasks\n",
+    "- Split the tomography problem into chunks along the angle axis.\n",
+    "- Solve the split problem using the [`odl.solvers.kaczmarz`](https://github.com/odlgroup/odl/blob/ad32a286b69f34260d4428d7282b4058ed2e2603/odl/solvers/iterative/iterative.py#L387-L517) solver.\n",
+    "- Use a different block scheme (block-sequential, interlaced, maximizing inter-block angle distance etc.) for arranging the angles, and investigate the convergence behavior.\n",
+    "\n",
+    "Use  to solve a tomography problem by splitting along the angles. Experiment a bit with blocking schemes .\n",
     "\n",
     "### Good to know\n",
     "\n",